Centrography

Central Feature

The most centrally located feature in a point, line, or polygon input

Directional Distribution

Mean Center

Mean Center

Standard Distance

Neighborhood Summary Statistics

Neighborhood Types

• Distance band
• Number of neighbors
• Contiguity edges only (also corners)
• Theissen tessalation

What is density

Density analysis takes known quantities of some phenomenon and spreads them across the landscape based on the quantity that is measured at each location and the spatial relationship of the locations of the measured quantities.

Kernel Density Estimates

Conceptually, a smoothly curved surface is fitted over each point. The surface value is highest at the location of the point and diminishes with increasing distance from the point, reaching zero at the Search radius distance from the point.

Example of Kernel Density Usage

Kernel density estimation is shown without a barrier (1) and with a barrier on both sides of the roads (2).

Point and Line Density

Complete Spatial Randomness

• Any event has equal probability of occurring in any location, a 1st order effect.
• The location of one event is independent of the location of another event, a 2nd order effect.

Could the distribution of Walmart stores in MA have been the result of a CSR process?

Point Patterns

AAN Tool

ArcGIS’ average nearest neighbor (ANN) tool computes the 1st nearest neighbor mean distance for all points. If ANN_ratio = 1, we have a random process. If ANN_ratio > 1, it is dispersed, and if ANN_ratio < 1 it is clustered.

ArcGIS Average Nearest Neighbor Tool (ANN)

ANN with specified extent

Hypothesis testing

• Set null hypothesis (the data is generated from CSR process)
• Simulate many realizations of the process and compute a statistic (ANN or other). Plot it on the histogram. Compare simulated values with observed values.
• Calculate \(p\)-value: Count the number of more extreme realizations and devide by the total number of realizations. \(\frac{319+1}{1000+1} = 0.32\)

Hypothesis testing II

Now imagine that we can take into account other factors (population density, median income) when generating random realizations of Walmart siting.

Expected versus observed

\(p\)-value is below 0.05, we can reject null hypothesis. Population density affects the placement of Walmart stores.